ASYMPTOTIC COHOMOLOGY OF CIRCULAR UNITS

by

Jean-Robert Belliard

Abstract. — Let F be a number field, abelian over ℚ, and fix a prime p ≠ 2. Consider the cyclotomic ℤ_{p}-extension F_{∞}/F and denote F_{n} the n^{th} finite subfield and C_{n} its group of circular units. Then the Galois groups G_{m,n} = Gal(F_{m}/F_{n}) act naturally on the C_{m}’s (for any m ≥ n >> 0). We compute the Tate cohomology groups Ĥ^{i}(G_{m,n}, C_{m}) for i = −1, 0 without assuming anything else neither on F nor on p.

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1. Two versions of cyclotomic units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2. Universal norms of circular units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3. Cohomology of circular units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4. Universal co-norms of circular units . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Introduction

Let F be a number field, abelian over ℚ, and fix a prime p ≠ 2. Consider the cyclotomic ℤ_{p}-extension F_{∞}/F and denote F_{n} its n^{th} finite layer, i.e. F_{n}/F is the unique sub-extension of degree p^{n} of F_{∞}/F. Following Sinnott ([S]) we define, for every F_{n}, its group of circular units C_{n}. Then the cyclic Galois groups G_{m,n} = Gal(F_{m}/F_{n}) act naturally on the C_{m}’s (for any m ≥ n ≥ 0). The present note aims to compute the Tate cohomology groups Ĥ^{i}(G_{m,n}, C_{m}) for i = −1, 0 without assuming anything else on F. To complete our computations in this general case, we need to take into account differences between Sinnott’s and Washington’s versions of circular units, even at the infinite level. In the process, as it does not involve any other difficulties, we compute the cohomology of this second version, and also the Γ_{n} = Gal(F_{∞}/F_{n})-cohomology of both inductive limits. Unfortunately we only succeed in describing these cohomology groups by assuming that n is not less than some natural number N (see lemma 2.8, lemma 3.1 and theorem 3.5), hence the word ”asymptotic” in the title. On the other hand if we do assume ad hoc hypotheses on F, then the natural number N may be chosen to be equal to 0, and our general approach recovers all previously known special cases (see e.g. articles [K1, K2, K3, KO], and §III.4 of [B1], and §4 of [N]).

2000 Mathematics Subject Classification. — 11R23,11R18,11R34.

This paper was written while I was hosted by the CNRS at the Institut de Mathématiques de Bordeaux. I thank both these institutions for their hospitality.

1. Two versions of cyclotomic units

We start by recalling the definitions of Sinnott’s and Washington’s circular units. Fix an embedding of \overline{ℚ} in ℂ and denote ζ_{m} = exp(2iπ/m).

Definition 1.1. — Let F be an abelian number field and f be its conductor, that is the smallest integer such that F ⊂ ℚ(ζ_{f}). The group of units of F will be denoted U(F).

1. We call group of circular numbers of F and denote Cyc(F) the Galois submodule of F^{×} generated by −1 and the numbers N_{ℚ(ζ_{d})/ℚ(ζ_{d})∩F}(1 − ζ_{d}), where d runs through all divisors of f greater than 1.

2. We call Sinnott’s group of circular units of F (see [S]) and denote C(F) the

intersection

C(F) = Cyc(F) ⋂ U(F).

3. We call Washington’s group of circular units of F (see p.143 of [W]) and denote W (F) the intersection

W (F) = Cyc(ℚ(ζ_{f})) ⋂ U(F).

Fix now a prime number p ≠ 2 and consider the cyclotomic ℤ_{p}-extension F_{∞}/F of \underline{any abelian} number field F. Then all finite layers F_{n} are still abelian over ℚ and the notations C_{n} = C(F_{n}) and W_{n} = W (F_{n}) make sense. We also abbreviate U(F_{n}) to U_{n}. When necessary we will rather consider the pro-p-completions \overline{C}(F) = C(F) ⊗ ℤ_{p}, \overline{W}_{n} = W_{n} ⊗ ℤ_{p}, \overline{C}_{n} = C_{n} ⊗ ℤ_{p} and \overline{U}_{n} = U_{n} ⊗ ℤ_{p}. We put as usual Γ = Gal(F_{∞}/F) and denote Λ = ℤ_{p}[[Γ]] for the Iwasawa algebra. For all n we have obvious inclusions C_{n} ⊂ W_{n}. Counterexamples to the equality are known. Moreover there exist also abelian number fields F such that even the projective limits \overline{C}_{∞} = lim_{←} \overline{C}_{n} and \overline{W}_{∞} = lim_{←} \overline{W}_{n} with respect to norm maps along F_{∞}/F disagree (see e.g. [B2]; other examples were built simultaneously and independently by Kučera in [Kuč]).

Definition 1.2. — Let us denote KN_{∞} the quotient module KN_{∞} = \overline{W}_{∞}/\overline{C}_{∞}. This module has been defined as a capitulation kernel for unit classes and called ”Kučera–Nekovář kernel” in [BN] and [NL].

Let \overline{U}_{∞} be the projective limit of the \overline{U}_{n}. Let \overline{μ}_{n} be the p^{th}-power roots of unity in F_{n} and \overline{μ}_{∞} be their projective limit. Let F^{+} be the maximal real subfield of F, and for any p-adic Gal(F/F^{+})-module M let us abbreviate M^{Gal(F/F^{+})} to M^{+}. We have \overline{μ}_{∞} ⊂ \overline{C}_{∞} ⊂ \overline{W}_{∞} ⊂ \overline{U}_{∞} and with M standing for any of these four modules of units we have a direct sum decomposition M = M^{+} ⊕ \overline{μ}_{∞}. For later use, we summarize some other known properties of these modules in a proposition.

Proposition 1.3. — Let r = [F^{+} : ℚ].

1. The module KN_{∞} is finite.

2. The module \overline{W} _{∞}^{+} is Λ-free of rank r.

3. The Λ-module \overline{C}_{∞}^{+} is of rank r.

4. The module \overline{C}_{∞}^{+} is Λ-free if and only if KN_{∞} = 0.

5. The module KN_{∞} is the maximal finite submodule of \overline{U}_{∞}/\overline{C}_{∞} = \overline{U}_{∞}^{+}/\overline{C}_{∞}^{+}.

Proof. — The indices (W_{n} : C_{n}) are bounded uniformly in n, this is the main result of [KN]. Assertion 1 follows. Assertion 2 is a theorem of Kuz′min (see [Kuz2] or discussion before proposition 3.6 of [B2]). Assertion 3 is a consequence of 1 and 2. Assertion 4 is proposition 3.6 of [B2] (see also [BN], proposition 2.3 (ii)). Let us denote MF_{∞} for the maximal finite submodule of \overline{U}_{∞}/\overline{C}_{∞}. By assertion 1 we already have an inclusion KN_{∞} ⊂ MF_{∞}. But since both \overline{U}_{∞}^{+} and \overline{W} _{∞}^{+} are Λ-free the quotient \overline{U}_{∞}/\overline{W}_{∞} has no non-trivial finite submodule. This shows assertion 5. □

In [BN] (proposition 2.3 (ii)) the module denoted KN_{F} was shown to be equal to MF_{∞}, so that notations are coherent. For our present purpose we will make no use of the interpretation in terms of capitulation kernels. In the sequel we will consider this finite module KN_{∞} as a parameter depending on the base field F. As written above, two different infinite families of couples (F, p) giving non-trivial KN_{∞} are described in [Kuč] and [B2]. On the other hand the definition 6.4, together with theorem 6.6 in the appendix to [NL] gives a criterion for the triviality of KN_{∞}. Up to now all known examples of trivial KN_{∞} satisfy this criterion.

2. Universal norms of circular units

To compute the cohomology of C_{m} and W_{m} we first use the simpler structure of \overline{W} _{∞}^{+} and of \overline{C}_{∞}^{+} to recover the cohomology of the ”universal norms” modules C̃_{n}^{+} ⊂ \overline{C}_{n} and W̃_{n}^{+} ⊂ \overline{W}_{n}, which are defined as the images of the usual projection maps. Then in section 3 we will control the deviation between the initial modules and their universal norms.

Definition 2.1. — Let n ∈ ℕ.

1. Let C̃_{n} = Im(\overline{C}_{∞} ⟶ \overline{C}_{n}) be the module consisting of the universal norms of Sinnott’s units.

2. Let W̃_{n} = Im(\overline{W}_{∞} ⟶ \overline{W}_{n}) be the module consisting of the universal norms of Washington’s units.

By an usual compactness argument we have e.g. C̃_{n} = ⋂_{m≥n} N_{F_{m}/F_{n}}(\overline{C}_{m}) hence the terminology ”universal norms”.

In the sequel we will obtain asymptotic results but will try to collect as much information as possible about the first layer F_{n} from which our results apply. For the cohomology of W̃_{n}, this is easier. It is well known that there exists some integer n such that F_{∞}/F_{n} is totally ramified at every place above p.

Definition 2.2. — We will denote n_{d} the smallest integer such that no place above p do splits anymore in F_{∞}/F_{n_{d}}.

If p is (at most) tamely ramified in F/ℚ, then n_{d} = 0. For all m ≥ n ≥ 0 we will denote Gal(F_{∞}/F_{n}) by Γ_{n} and Gal(F_{m}/F_{n}) ≅ Γ_{n}/Γ_{m} by G_{m,n}. By the mere definition of Galois action on places, the group Γ_{n_{d}} acts trivially on the set S_{∞} of places of F_{∞} dividing p.

For later use we will isolate in a lemma here a precise statement for our context of the well known snake lemma :

Lemma 2.3. — For all m ∈ ℕ and all Λ-module M let us denote by M^{Γ_{m}} the Γ_{m}-invariant submodule of M and by M_{Γ_{m}} the Γ_{m}-coinvariant quotient module of M. Let 0 → A → B →^{φ} C → 0 be an exact sequence of Λ-module. Pick for all m a topological generator γ_{m} of Γ_{m}. Then for all m ∈ ℕ the map δ: C^{Γ_{m}} ⟶ A_{Γ_{m}} is well defined by the formula δ(φ(b)) = (γ_{m} − 1)b + (γ_{m} − 1)A and fits into the exact sequence

0 ⟶ A^{Γ_{m}} ⟶ B^{Γ_{m}} ⟶ C^{Γ_{m}} ⟶^{δ} A_{Γ_{m}} ⟶ B_{Γ_{m}} ⟶ C_{Γ_{m}} ⟶ 0.

Proof. — Just apply the snake lemma to the sequence 0 → A → B →^{φ} C → 0 connected with itself by multiplication by γ_{m} − 1. □

Proposition 2.4. — Let s^{+} = #S_{∞}^{+} be the number of places of F_{∞}^{+} dividing p. Let n_{d} be as in definition 2.2. Then for all m ≥ n ≥ n_{d}, we have Ĥ^{0}(G_{m,n}, W̃_{m}) = 0 and H^{1}(G_{m,n}, W̃_{m}) is free of rank s^{+} over ℤ/p^{m−n}ℤ.

Proof. — Fix m ≥ n ≥ n_{d}. The canonical surjection \overline{W}_{∞} ⟶ W̃_{m} factors through w_{m}: (\overline{W}_{∞})_{Γ_{m}} ↠ W̃_{m}. Set T_{m} = Ker w_{m}. By proposition 1.3 we know that (\overline{W}_{∞})_{Γ_{m}} is isomorphic to \overline{μ}_{m} ⊕ (\overline{W} _{∞}^{+})_{Γ_{m}} with both summands G_{m,0}-cohomologically trivial. Therefore the cohomology of W̃_{m} is the cohomology of T_{m} shifted once. To conclude the proof of proposition 2.4 it suffices to show that T_{m} is a free ℤ_{p}-module of rank s^{+} with trivial action of G_{m,n}. For all n let \overline{U}′_{n} denote the pro-p-completion of (p)- units of F_{n}, and \overline{U}′_{∞} be their projective limit. By two theorems of Kuz′min (see theorems 7.2 and 7.3 in [Kuz1]) the module \overline{U}′_{∞} is Λ-free and for all m the map (\overline{U}′_{∞})_{Γ_{m}} ⟶ \overline{U}′_{m} is injective.

The sequence \xymatrix{0 \ar[r] & \overline{W}_{∞} \ar[r] & \overline{U}′_{∞} \ar[r] & \overline{U}′_{∞}/\overline{W}_{∞} \ar[r] & 0} gives by lemma 2.3

\xymatrix{0 \ar[r] & (\overline{U}′_{∞}/\overline{W}_{∞})^{Γ_{m}} \ar[r] & (\overline{W}_{∞})_{Γ_{m}} \ar[r] & (\overline{U}′_{∞})_{Γ_{m}}} hence an isomorphism

T_{m} ≅ (\overline{U}′_{∞}/\overline{W}_{∞})^{Γ_{m}}.

This, together with the following lemma 2.5, completes the proof of the proposition 2.4. □

Lemma 2.5. — Let s_{m}^{+} be the cardinality of the set S_{m}^{+} of places dividing p in the maximal real subfield F_{m}^{+} of F_{m}. Note that for all m ≥ n_{d} the set S_{m}^{+} is in bijection with S_{∞}^{+}.

1. The module \overline{U}′_{∞}/\overline{U}_{∞} is ℤ_{p}-free and pseudo-isomorphic to ℤ_{p}[S_{∞}^{+}].

2. For all m the module (\overline{U}′_{∞}/\overline{W}_{∞})^{Γ_{m}} is ℤ_{p}-free of rank s_{m}^{+} with trivial action by Γ_{n_{d}}.

Proof. — We first prove that 1 implies 2, then we will prove 1. Apply lemma 2.3 to the sequence

\xymatrix{0 \ar[r] & \overline{U}_{∞}/\overline{W}_{∞} \ar[r] & \overline{U}′_{∞}/\overline{W}_{∞} \ar[r] & \overline{U}′_{∞}/\overline{U}_{∞} \ar[r] & 0.}

We get

\xymatrix{0 \ar[r] & (\overline{U}_{∞}/\overline{W}_{∞})^{Γ_{m}} \ar[r] & (\overline{U}′_{∞}/\overline{W}_{∞})^{Γ_{m}} \ar[r] & (\overline{U}′_{∞}/\overline{U}_{∞})^{Γ_{m}} \ar[r] & (\overline{U}_{∞}/\overline{W}_{∞})_{Γ_{m}}}

By lemma 1.3 the module \overline{U}_{∞}/\overline{W}_{∞} is pseudo-isomorphic to \overline{U}_{∞}/\overline{C}_{∞}, and as a consequence of Coleman’s theory and Leopoldt’s conjecture (see theorem 1.1 of [B3]) these two modules have finite Γ_{m}-invariants and co-invariants. But since \overline{U}_{∞}^{+} and \overline{W} _{∞}^{+} are Λ-free the quotient \overline{U}_{∞}/\overline{W}_{∞} has no non-trivial finite submodule so that (\overline{U}_{∞}/\overline{W}_{∞})^{Γ_{m}} = 0 and our previous sequence becomes

\xymatrix{0 \ar[r] & (\overline{U}′_{∞}/\overline{W}_{∞})^{Γ_{m}} \ar[r] & (\overline{U}′_{∞}/\overline{U}_{∞})^{Γ_{m}} \ar[r] & finite.}

Therefore assertion 2 follows from assertion 1. To prove 1 we use (normalized) valuations at places above p and consider the exact sequence :

\xymatrix{0 \ar[r] & \overline{U}′_{∞}/\overline{U}_{∞} \ar[r]^{val} & ℤ_{p}[S_{∞}] \ar[r] & D_{∞} \ar[r] & 0,}

where D_{∞} stands for (the projective limit) of the p-part of the subgroup of ideal class groups generated by places above p. This shows already that (\overline{U}′_{∞}/\overline{U}_{∞}) is ℤ_{p}-free with trivial action of Γ_{n_{d}}. Proving that (\overline{U}′_{∞}/\overline{U}_{∞}) is pseudo-isomorphic to ℤ_{p}[S_{∞}^{+}] is then equivalent to prove that D_{∞} is pseudo-isomorphic to ℤ_{p}[S_{∞}]^{−} = ℤ_{p}[S_{∞}]/ℤ_{p}[S_{∞}^{+}]. This last statement is a consequence of Leopoldt’s conjecture (which holds true for our abelian field F) and is part of the folklore. For instance Greenberg in [G] §1 proves that D_{∞}^{+} is finite and constructs in loc. cit. §2 a sequence of subgroups (denoted C_{m} in loc. cit. p.120) of uniformly bounded finite index in D_{m}, whose projective limit is pseudo-isomorphic to ℤ_{p}[S_{∞}]^{−}. □

Corollary 2.6. — Let W̃_{→∞} = lim_{→}_{m}W̃_{m} be the inductive limit of the W̃_{m}’s. For all

n ≥ n_{d} we have a (group)-isomorphism H^{1}(Γ_{n}, W̃_{→∞}) ≃ (ℚ_{p}/ℤ_{p})^{s^{+}} and H^{2}(Γ_{n}, W̃_{→∞}) is trivial.

Proof. — The groups H^{i}(Γ_{n}, W̃_{→∞}) are isomorphic to the inductive limit with respect

to the maps H^{i}(G_{m,n}, W̃_{m}) ⟶ H^{i}(G_{m+1,n}, W̃_{m+1}) induced by the couples of maps G_{m+1,n} → G_{m,n} and W̃_{m} → W̃_{m+1} (see proposition 1.5.1 of [NSW]). By proposition 2.4 the W̃_{m} satisfies Galois descent, therefore the inflation maps H^{1}(G_{m,n}, W̃_{m}) ⟶ H^{1}(G_{m+1,n}, W̃_{m+1}) are injectives. As H^{1}(G_{m,n}, W̃_{m}) ≃ (ℤ/p^{m−n})^{s^{+}} the first limit is (ℚ_{p}/ℤ_{p})^{s^{+}}. Triviality of all H^{2} ’s follows from that of all the Ĥ^{0}’s. □

We want to describe the G_{m,n}-cohomology of the C̃_{m}’s and we will use a variation on the above method used for the W̃_{m}’s. As at the beginning of the proof of proposition 2.4 we have an isomorphism Ker((\overline{C}_{∞})_{Γ_{m}} ⟶ C̃_{m}) ≅ (\overline{U}′_{∞}/\overline{C}_{∞})^{Γ_{m}}. But \overline{U}′_{∞}/\overline{C}_{∞} may have a non trivial finite submodule. Indeed we have sequence

\xymatrix{0 \ar[r] & KN_{∞}^{Γ_{m}} \ar[r] & (\overline{U}′_{∞}/\overline{C}_{∞})^{Γ_{m}} \ar[r] & (\overline{U}′_{∞}/\overline{W}_{∞})^{Γ_{m}} \ar[r] & finite,} (1)

extracted from the lemma 2.3 applied to the sequence 0 → KN_{∞} → \overline{U}′_{∞}/\overline{C}_{∞} → \overline{U}′_{∞}/\overline{W}_{∞} → 0.

Definition 2.7. — We define V_{m} := (\overline{U}′_{∞}/\overline{C}_{∞})^{Γ_{m}}/KN_{∞}^{Γ_{m}}.

From the sequence (1) and lemma 2.5 we see that V_{m} is ℤ_{p}-free of rank s^{+} and the natural action by Γ_{n_{d}} on V_{m} is trivial. Hence for n ≥ n_{d} the G_{m,n}-cohomology of V_{m} is just s^{+} copies of the cohomology of ℤ. We will get the cohomology of C̃_{m} by going through the cohomology of the sequence :

\xymatrix{0 \ar[r] & V_{m} \ar[r] & (\overline{C}_{∞})_{Γ_{m}}/KN_{∞}^{Γ_{m}} \ar[r] & C̃_{m} \ar[r] & 0} (2)

Before this computation we will prove the existence of a first layer n from which our result will apply and try to make it as precise as possible.

Lemma 2.8. — 1. There exists an n such that for all integers m ≥ n we have

(\overline{U}′_{∞}/\overline{C}_{∞})^{Γ_{m}} = (\overline{U}′_{∞}/\overline{C}_{∞})^{Γ_{n}} .

2. For any integer n satisfying 1, the group Γ_{n} acts trivially on KN_{∞}.

3. If n satisfies 1, then n ≥ n_{d}.

4. If KN_{∞} = 0 then the integer n_{d} of definition 2.2 satisfies 1.

Proof. — The Λ-module \overline{U}′_{∞}/\overline{C}_{∞} is finitely generated. The sequence of submodules ((\overline{U}′_{∞}/\overline{C}_{∞})^{Γ_{m}})_{m∈ℕ} is increasing. Part 1 of the lemma follows by noetherianity. Part

2 comes from the inclusion KN_{∞} ⊂ \overline{U}′_{∞}/\overline{C}_{∞}. Suppose that some x ∈ KN_{∞} is such that σx ≠ x for some σ ∈ Γ_{n}. But x, as an element of the finite module KN_{∞}, must be fixed by some Γ_{m} for m > n. Hence x ∈ (\overline{U}′_{∞}/\overline{C}_{∞})^{Γ_{m}} \ (\overline{U}′_{∞}/\overline{C}_{∞})^{Γ_{n}} and n does not satisfy 1. This shows assertion 2. For assertion 3, and 4 we have seen in the proof of lemma 2.5 that (\overline{U}′_{∞}/\overline{W}_{∞})^{Γ_{m}} embeds with finite cokernel into ℤ_{p}[S_{∞}] for all m ≥ n_{d}. Now Γ_{n_{d}} acts trivially on ℤ_{p}[S_{∞}] and if n_{d} > 0 then ℤ_{p}[S_{∞}]^{Γ_{n_{d}−1}} is not of finite index in ℤ_{p}[S_{∞}]. This proves 3 and 4. □

Definition 2.9. — We will denote n_{U′C} the smallest integer n such that for all integers m ≥ n the equality (\overline{U}′_{∞}/\overline{C}_{∞})^{Γ_{m}} = (\overline{U}′_{∞}/\overline{C}_{∞})^{Γ_{n}} holds.

This number n_{U′C} is the first nonexplicit part of our ”asymptotic” approach. Other approaches found in the literature simply assume hypotheses that imply KN_{∞} = 0 and in such case we simply have n_{U′C} = n_{d}.

Theorem 2.10. — Let s^{+} be as in proposition 2.4. Let n be greater than or equal to n_{U′C}. Let V_{m} be the ℤ_{p}-free module of rank s^{+} and trivial Γ_{n_{d}}-action of definition 2.7. For every integer i let KN_{∞}[p^{i}] be the kernel of multiplication by p^{i} in KN_{∞}. For every m ≥ n we have

Ĥ^{0}(G_{m,n}, C̃_{m}) ≅ KN_{∞}[p^{m−n}]

and a (group split) exact sequence

\xymatrix{0 \ar[r] & KN_{∞}/p^{m−n} \ar[r] & H^{1}(G_{m,n}, C̃_{m}) \ar[r] & V_{m}/p^{m−n} \ar[r] & 0} .

Proof. — Fix m ≥ n ≥ n_{U′C}. To describe the cohomology of the sequence (2), we first compute the cohomology of (\overline{C}_{∞})_{Γ_{m}}/KN_{∞}. From the sequence 0 → \overline{C}_{∞} → \overline{W}_{∞} → KN_{∞} → 0 we obtain by lemma 2.3 a sequence

\xymatrix{0 \ar[r] & (\overline{C}_{∞})_{Γ_{m}}/KN_{∞} \ar[r] & (\overline{W}_{∞})_{Γ_{m}} \ar[r] & KN_{∞} \ar[r] & 0.}

As (\overline{W}_{∞})_{Γ_{m}} is G_{m,n}-cohomologicaly trivial we see that (\overline{C}_{∞})_{Γ_{m}}/KN_{∞} has the G_{m,n}- cohomology of KN_{∞} shifted once. The exact hexagone of cyclic Tate cohomology associated to the sequence (2) is now :

\xymatrix{V_{m}/p^{m−n} \ar[r]^{α} & KN_{∞}[p^{m−n}] \ar[r] & Ĥ^{0}(G_{m,n}, C̃_{m}) \ar[d] \\ Ĥ^{−1}(G_{m,n}, C̃_{m}) \ar[u] & KN_{∞}/p^{m−n} \ar[l] & 0 \ar[l]}

To conclude the proof of the theorem it remains now to prove that the map α of this diagram is the 0 map. This map α is induced by the map δ: (\overline{U}′_{∞}/\overline{C}_{∞})^{Γ_{m}} ⟶ (\overline{C}_{∞})_{Γ_{m}} of the lemma 2.3 applied to the sequence 0 → \overline{C}_{∞} → \overline{U}′_{∞} → \overline{U}′_{∞}/\overline{C}_{∞} → 0. To define this map δ let us pick a choice of a generator γ_{m} of Γ_{m}, and for later use let us denote γ_{n} the generator of Γ_{n} such that γ_{n}^{p^{m−n}} = γ_{m}. Then the map δ sends any coset \overline{u} ∈ \overline{U}′_{∞}/\overline{C}_{∞} such that (γ_{m} − 1)u ∈ \overline{C}_{∞} to the coset of (γ_{m} − 1)u in (\overline{C}_{∞})_{Γ_{m}}. Note that γ_{m} − 1 = ν_{m,n}(γ_{n} − 1), where ν_{m,n} = ∑_{i=0}^{p^{m−n}−1} γ_{n}^{i} ∈ Λ acts like the algebraic norm of the extension F_{m}/F_{n} on any Γ_{m}-trivial module. But the integers m ≥ n have been chosen in such a way that (\overline{U}′_{∞}/\overline{C}_{∞})^{Γ_{m}} = (\overline{U}′_{∞}/\overline{C}_{∞})^{Γ_{n}} and therefore we have (γ_{m} − 1)u = ν_{m,n}(γ_{n} − 1)u, where (γ_{n} − 1)u ∈ \overline{C}_{∞}. As by definition we have

Ĥ^{0}(G_{m,n}, (\overline{C}_{∞})_{Γ_{m}}/KN_{∞}) = \frac{((\overline{C}_{∞})_{Γ_{m}}/KN_{∞})^{G_{m,n}}}{ν_{m,n}((\overline{C}_{∞})_{Γ_{m}}/KN_{∞})},

this shows that α is indeed the 0 map and concludes the proof of theorem 2.10. □

Corollary 2.11. — Let C̃_{→∞} = lim_{→}_{m}C̃_{m} be the inductive limit of the C̃_{m}’s. For any

n ≥ n_{U′C} we have a (group)-isomorphism H^{1}(Γ_{n}, C̃_{→∞}) ≃ (ℚ_{p}/ℤ_{p})^{s^{+}} and H^{2}(Γ_{n}, C̃_{→∞}) is trivial.

Proof. — As Γ_{n} is pro-p-free, H^{2}(Γ_{n}, C̃_{→∞}) is divisible; and its order is bounded by the

finite order of KN_{∞}, hence H^{2}(Γ_{n}, C̃_{→∞}) is trivial. Let us compute the H^{1}(Γ_{n}, C̃_{→∞}). Extracted mutatis-mutandis from the proof of the theorem 2.10 we have the exact sequence for all m >> 0 :

0 ⟶ H^{1}(G_{m,n}, (\overline{C}_{∞})_{Γ_{m}}/KN_{∞}) ⟶ H^{1}(G_{m,n}, C̃_{m}) ⟶ H^{2}(G_{m,n}, V_{m}) ⟶ 0.

Of course we have for m >> 0 the isomorphisms H^{1}(G_{m,n}, (\overline{C}_{∞})_{Γ_{m}}/KN_{∞}) ≃ KN_{∞} and H^{2}(G_{m,n}, V_{m}) ≃ V_{m}/p^{m−n} and we recover the exact sequence :

0 ⟶ KN_{∞} ⟶ H^{1}(G_{m,n}, C̃_{m}) ⟶ V_{m}/p^{m−n} ⟶ 0.

We want to take inductive limits on this sequence, and for that we need to clarify what are the extension maps going up from the mth to the (m + 1)th step for the modules KN_{∞} and the modules V_{m}/p^{m−n}. The finite module of universal norms KN_{∞} stabilizes with respect to natural (going down) norm maps and composition with extension maps gives multiplication by p. Therefore the inductive limit with respect to extension maps of KN_{∞} is trivial. On the contrary the modules (V_{m})_{m} stabilize with respect to extension maps already from the first step to V_{n} for m ≥ n ≥ n_{U′C}. So for fixed n ≥ n_{U′C} the inductive limit lim_{→} V_{m}/p^{m−n} is V_{n} ⊗ ℚ_{p}/ℤ_{p} ≃

(ℚ_{p}/ℤ_{p})^{s^{+}}. □

3. Cohomology of circular units

We will try and recover the cohomology of \overline{C}_{m} and \overline{W}_{m} from that of their universal norms C̃_{m} and W̃_{m}. For \overline{C}_{m} the method is not new and has already been used in [N] or [NL] and originally in [B1]. Here we only encounter another slight difficulty coming from the non triviality of Ĥ^{0}(G_{m,n}, C̃_{m}). For \overline{W}_{m} our result is original.

Lemma 3.1. — For all n let I_{n} be the inertia subfield for p in F_{n}/ℚ. We have I_{n+1} = I_{n} as soon as F_{n+1}/F_{n} ramifies. Let I = ⋃_{n} I_{n} be the inertia subfield for p in F_{∞}/ℚ.

1. For all n we have equality

\overline{C}_{n} = C̃_{n}\overline{C}(I_{n}),

hence for all n such that F_{n+1}/F_{n} ramifies we have equality

\overline{C}_{n} = C̃_{n}\overline{C}(I).

2. There exist an integer n such that for all integers m ≥ n we have

C̃_{m} ⋂ C(I) = C̃_{n} ⋂ C(I).

Proof. — Assertion 1 is lemme 2.5 of [B2]. The ℤ_{p}-module C(I) is finitely generated. The sequence of submodules (C̃_{m} ⋂ C(I))_{m∈ℕ} is increasing. Assertion 2 follows by noetherianity. □

Definition 3.2. — We will denote n_{C} the smallest non negative integer such that F_{n_{C}+1}/F_{n} ramifies and such that property 2 of lemma 3.1 holds.

This number n_{C} is our second nonexplicit asymptotic constant. However if we assume KN_{∞} = 0 then theorem 2.10 shows that the C̃_{m}’s satisfy Galois descent and that proves the equality n_{C} = n_{i}, where n_{i} is the smallest non negative integer such that F_{n_{i}+1}/F_{n_{i}} ramifies.

After KN_{∞} there is another module that we have to consider as a parameter depending on F. This module is essentially the one denoted Φ in [N] and that we will define in our more general context now :

Definition 3.3. — We call universal co-norms of circular units and denote by Φ_{m} the quotient module

Φ_{m} = \overline{C}_{m}/C̃_{m}.

We state and prove now the properties of Φ_{m} which are needed to compute the cohomology of \overline{C}_{m}. In particular we fix its asymptotic behavior, so that we may consider it as a parameter (even if it is a non explicit and asymptotic one !). We will give somewhat more explicit informations about this module and examine some examples in section 4.

Lemma 3.4. — Recall that n_{i} is the first nonnegative integer such that F_{n_{i}+1}/F_{n_{i}} ramifies.

1. For all m the ℤ_{p}-rank of Φ_{m} is s_{m}^{+} − 1 where s_{m}^{+} is the cardinal of the set S_{m}^{+} of places dividing p in the maximal real subfield F_{m}^{+} of F_{m}.

2. For all m the group Γ_{n_{i}} acts trivially on Φ_{m}.

3. For any m ≥ n_{C} the extension map \overline{C}_{n_{C}} ⟶ \overline{C}_{m} induces an isomorphism

Φ_{n_{C}} ≅ Φ_{m}.

Proof. — This lemma is an easy generalization of lemma-definition 2.2 in [NL], but for the convenience of the reader we reprove it. To compute the rank of Φ_{m} just use the exact sequence

\xymatrix{0 \ar[r] & (\overline{U}′_{∞}/\overline{C}_{∞})^{Γ_{m}} \ar[r] & (\overline{C}_{∞})_{Γ_{m}} \ar[r] & \overline{C}_{m} \ar[r] & Φ_{m} \ar[r] & 0} .

Let r = [F^{+} : ℚ]. Then the Λ-module \overline{C}_{∞} has trivial Γ-invariants and rank r. It follows that (\overline{C}_{∞})_{Γ_{m}} has ℤ_{p}-rank equal to rp^{m}. Also the ℤ_{p}-module \overline{C}_{m} is of finite index in \overline{U}_{m} and has rank rp^{m} − 1 by Dirichlet’s theorem. The module (\overline{U}′_{∞}/\overline{C}_{∞})^{Γ_{m}} has the same rank as (\overline{U}′_{∞}/\overline{W}_{∞})^{Γ_{m}} i.e. s_{m}^{+} by assertion 2 of lemma 2.5. This proves 1. Assertions 2 and 3 follow from lemma 3.1. Indeed for all m we have

Φ_{m} = \overline{C}_{m}/C̃_{m} = C̃_{m}C(I_{m})/C̃_{m} ≅ C(I_{m})/C̃_{m} ⋂ C(I_{m}).

For all m the action of Γ_{n_{i}} is trivial on I_{m}, which proves 2 and for all m ≥ n_{C} the extension map Φ_{n_{C}} ⟶ Φ_{m} commutes with the identity map in

\overline{C}(I)/(C̃_{n_{C}} ∩ \overline{C}(I)) = \overline{C}(I)/(C̃_{m} ∩ \overline{C}(I)),

which proves 3. □

In the sequel we will abbreviate Φ_{n_{C}} to simply Φ. Recall that from theorem 2.10 we have for all m ≥ n ≥ n_{U′C} isomorphisms Ĥ^{0}(G_{m,n}, C̃_{m}) ≅ KN_{∞}[p^{m−n}] and exact sequences :

\xymatrix{0 \ar[r] & KN_{∞}/p^{m−n}KN_{∞} \ar[r] & Ĥ^{−1}(G_{m,n}, C̃_{m}) \ar[r] & (ℤ/p^{m−n})^{s^{+}} \ar[r] & 0} .

Theorem 3.5. — Let n be any integer not less than both n_{U′C} and n_{C}. Then for all m ≥ n we have exact sequences

\xymatrix{0 \ar[r] & KN_{∞}[p^{m−n}] \ar[r] & Ĥ^{0}(G_{m,n}, \overline{C}_{m}) \ar[r] & Φ/p^{m−n}Φ \ar[r] & 0}

and

\xymatrix{0 \ar[r] & H^{1}(G_{m,n}, C̃_{m}) \ar[r] & H^{1}(G_{m,n}, \overline{C}_{m}) \ar[r] & Φ[p^{m−n}] \ar[r] & 0} .

Proof. — Fix m ≥ n with n ≥ max(n_{C}, n_{U′C}). As G_{m,n} is cyclic we may and will compute Ĥ^{−1} during the proof. This proof consists in splitting into two pieces the exact hexagone of cyclic Tate cohomology groups associated to the exact sequence 0 ⟶ C̃_{m} ⟶ \overline{C}_{m} ⟶ Φ ⟶ 0. At first sight this exact hexagone is

\xymatrix{KN_{∞}[p^{m−n}] \ar[r]^{φ_{0}} & Ĥ^{0}(G_{m,n}, \overline{C}_{m}) \ar[r] & Φ/p^{m−n}Φ \ar[d]^{δ_{0}} \\ Φ[p^{m−n}] \ar[u]^{δ_{−1}} & Ĥ^{−1}(G_{m,n}, \overline{C}_{m}) \ar[l] & Ĥ^{−1}(G_{m,n}, C̃_{m}). \ar[l]^{φ_{−1}}}

So we only have to prove that the two δ maps are both the 0 map or equivalently that the φ maps are injective. The injectivity of φ_{−1} follows from the stabilization with respect to extension maps of the quotients Φ_{m}. Indeed from this stabilization property we can write \overline{C}_{m} = \overline{C}_{n}C̃_{m} and therefore for all generator σ of G_{m,n} we get (σ − 1)(\overline{C}_{m}) = (σ − 1)(C̃_{m}) which proves that φ_{−1} is injective.

The kernel of φ_{0} is by definition the quotient N_{m,n}(\overline{C}_{m}) ∩ C̃_{m}/N_{m,n}(C̃_{m}), where N_{m,n} stands for the norm map of F_{m}/F_{n}. But by assertion 2 of lemma 3.1 we have

N_{m,n}(\overline{C}_{m}) = N_{m,n}(C̃_{m}\overline{C}(I)) = N_{m,n}(C̃_{m})\overline{C}(I)^{p^{m−n}}.

Hence the equalities

Ker φ_{0} = \frac{(N_{m,n}(C̃_{m})\overline{C}(I)^{p^{m−n}}) ⋂ C̃_{m}}{N_{m,n}(C̃_{m})} = \frac{N_{m,n}(C̃_{m}) (\overline{C}(I)^{p^{m−n}} ⋂ C̃_{m})}{N_{m,n}(C̃_{m})}.

But by the definition of n_{C} for m ≥ n ≥ n_{C} we have \overline{C}(I) ∩ C̃_{m} = \overline{C}(I) ∩ C̃_{n} and by definition of C̃_{n} we have C̃_{n} ⊂ N_{m,n}(C̃_{m}). This proves that Ker φ_{0} = 0. □

Corollary 3.6. — Let C^{→}_{∞} be the inductive limit with respect to extension maps of the \overline{C}_{m}’s. Let Tor(Φ) be the ℤ_{p}-torsion of Φ. For all n not less than both n_{C} and n_{U′C} we have (group) isomorphisms :

H^{1}(Γ_{n}, C^{→}_{∞}) ≃ (ℚ_{p}/ℤ_{p})^{s^{+}} ⊕ Tor(Φ),

and

H^{2}(Γ_{n}, C^{→}_{∞}) ≃ (ℚ_{p}/ℤ_{p})^{s^{+}−1}.

Proof. — As before, when taking inductive limits, all contributions from the module KN_{∞} vanish. The module Φ, contrary to KN_{∞}, stabilizes with respect to extension maps. Therefore extensions for Φ[p^{k}] ⟶ Φ[p^{k+1}] ultimately are isomorphisms and give Tor(Φ) as inductive limit, while extensions for Φ/p^{k} ⟶ Φ/p^{k+1} are right from the start induced by multiplication by p and give Φ ⊗ (ℚ_{p}/ℤ_{p}) ≃ (ℚ_{p}/ℤ_{p})^{s^{+}−1} as inductive limit. The sequence with H^{1} is group split because (ℚ_{p}/ℤ_{p})^{s^{+}} is an injective group. □

Finally we want to describe the cohomology of the W_{m}’s. We use the same strategy as for the C_{m}’s and for that, we first have to prove the stabilization by extension maps of the \overline{W}_{m}/W̃_{m}’s. We will deduce this stabilization from the one of the Φ_{m}’s, which unfortunately gives us a even worst lower bound n_{W} from which our results apply.

Lemma 3.7. — There Exists an n such that for any m ≥ n the extension map \overline{W}_{n} ⟶ \overline{W}_{m} induces an isomorphism \overline{W}_{n}/W̃_{n} ≅ \overline{W}_{m}/W̃_{m}.

Proof. — First take an n ≥ n_{C}, and consider the (tautological) diagram with exact rows :

\xymatrix{0 \ar[r] & (W̃_{n} ∩ \overline{C}_{n})/C̃_{n} \ar[r] & \overline{C}_{n}/C̃_{n} \ar[rr] \ar@{->>}[dr] &  & \overline{W}_{n}/W̃_{n} \ar[r] & \overline{W}_{n}/(\overline{C}_{n}W̃_{n}) \ar[r] & 0 \\  & (†) &  & \overline{C}_{n}/(W̃_{n} ∩ \overline{C}_{n}) \ar@{^{(}->}[ur]}

As n ≥ n_{C} the extension map Φ_{n} ⟶ Φ_{m} is surjective and so is the extension map \overline{C}_{n}/(W̃_{n} ∩ \overline{C}_{n}) ⟶ \overline{C}_{m}/(W̃_{n} ∩ \overline{C}_{m}). Recall that we have n ≥ n_{C} ≥ n_{i} ≥ n_{d}, so that theorem 2.4 applies. In particular the W̃_{m}’s satisfy Galois descent for m ≥ n and the extension map \overline{W}_{n}/W̃_{n} ⟶ \overline{W}_{m}/W̃_{m} is injective. Now use the snake lemma on the right hand part of the diagram (†) with extension maps. This gives :

\xymatrix{ &  & 0 \ar[d] \ar[r] & K_{n,m} \ar[d] \\ 0 \ar[r] & \overline{C}_{n}/(W̃_{n} ∩ \overline{C}_{n}) \ar[r] \ar[d] & \overline{W}_{n}/W̃_{n} \ar[r] \ar[d] & \overline{W}_{n}/(\overline{C}_{n}W̃_{n}) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \overline{C}_{m}/(W̃_{m} ∩ \overline{C}_{m}) \ar[r] \ar[d] & \overline{W}_{m}/W̃_{m} \ar[r] \ar[d] & \overline{W}_{m}/(\overline{C}_{m}W̃_{m}) \ar[r] \ar[d] & 0 \\  & 0 \ar[r] & CoK_{n,m} \ar[r] & CoK′_{n,m} \ar[r] & 0.}

It follows that the kernels K_{n,m} are trivial and that the co-kernels CoK_{n,m} and CoK′_{n,m} are isomorphic. By the triviality of the kernels we see that the orders of the finite groups \overline{W}_{n}/\overline{C}_{n}W̃_{n} are increasing with n, but must stabilize because they are uniformly bounded by the maximal of the orders of \overline{C}_{n}/\overline{W}_{n} (which exists according to main result of [KN]). So for an n greater than n_{C} and such that the order of \overline{W}_{n}/\overline{C}_{n}W̃_{n} is maximal, we get the triviality of the two cokernels CoK_{n,m}. □

Definition 3.8. — We will denote by n_{W} the smallest integer not less than n_{d} and such that for all m ≥ n the extension map \overline{W}_{n}/W̃_{n} ⟶ \overline{W}_{m}/W̃_{m} is an isomorphism. We will abbreviate \overline{W}_{m}/W̃_{m} to ΦW_{m} and \overline{W}_{n_{W}} /W̃_{n_{W}} to simply ΦW.

Using diagram (†) again we see that ΦW_{m} and Φ_{m} have the same ℤ_{p}-ranks, namely s_{m}^{+} − 1, and of course that the natural action of Γ_{n_{W}} on all ΦW_{m}’s is trivial.

Theorem 3.9. — For all m ≥ n ≥ n_{W} we have isomorphisms :

Ĥ^{0}(G_{m,n}, \overline{W}_{m}) ≅ ΦW/p^{m−n}

and (group split) exact sequences :

\xymatrix{0 \ar[r] & (ℤ/p^{m−n})^{s^{+}} \ar[r] & H^{1}(G_{m,n}, \overline{W}_{m}) \ar[r] & ΦW [p^{m−n}] \ar[r] & 0}

Proof. — Using lemma 3.7 the proof of this theorem is the same as the proof of theorem 3.5. It is even actually easier because when splitting in two pieces the exact hexagone associated to the sequence 0 ⟶ W̃_{m} ⟶ \overline{W}_{m} ⟶ ΦW ⟶ 0, we only have to prove the injectivity of the map corresponding to φ_{−1} by virtue of the triviality of Ĥ^{0}(G_{m,n}, W̃_{m}). This injectivity in turn follows from the stabilization of ΦW_{m} with respect to extension maps. □

Corollary 3.10. — Let W^{→}_{∞} be the inductive limit with respect to extension maps of the \overline{W}_{m}. Let Tor(ΦW) be the ℤ_{p}-torsion of ΦW. For all n greater than or equal to n_{W} we have (group) isomorphisms :

H^{1}(Γ_{n}, W^{→}_{∞}) ≃ (ℚ_{p}/ℤ_{p})^{s^{+}} ⊕ Tor(ΦW),

and

H^{2}(Γ_{n}, W^{→}_{∞}) ≃ (ℚ_{p}/ℤ_{p})^{s^{+}−1}.

Proof. — The proof is exactly the same as the one of corollary 3.6. □

4. Universal co-norms of circular units

In this section we want to investigate the module Φ_{m} = \overline{C}_{m}/C̃_{m} which, together with KN_{∞}, describes the cohomology of the \overline{C}_{m}’s. By lemma 3.4 we already know that its ℤ_{p}-rank is s_{m}^{+} − 1 where s_{m}^{+} is the number of places of F_{m}^{+} dividing p. As we have Φ_{m} = \overline{C}_{m}/C̃_{m} = \overline{C}_{m}^{+}/C̃_{m}^{+} we may suppose without loss of generality that F is totally real. Next, as the rank of Φ_{m} is known, we should concentrate on its ℤ_{p}-torsion, let’s say Tor(Φ_{m}). To give examples of non trivial Tor(Φ_{m}) we state and prove a lemma which provides also criteria for triviality of Tor(Φ_{m}).

Lemma 4.1. — Recall that I_{m} is the maximal subfield of F_{m} such that p does not ramify in I_{m}/ℚ, and that \overline{Cyc}(I_{m}) is the p-completion of its module of cyclotomic numbers. Let σ_{p} be the Fröbenius automorphism of I_{m}.

1. We have an exact sequence of finite groups

0 ⟶ C̃_{m} ∩ \overline{C}(I_{m})/\overline{Cyc}(I_{m})^{σ_{p}−1} ⟶ Ĥ^{−1}(⟨σ_{p}⟩, \overline{Cyc}(I_{m})) ⟶ Tor(Φ_{m}) ⟶ 0.

2. Assume that KN_{∞} = 0 and that p does not ramify in F, then we have an isomorphism Tor(Φ_{m}) ≅ Ĥ^{−1}(⟨σ_{p}⟩, \overline{Cyc}(I_{m}))

Proof. — By lemma 3.1 we have an isomorphism Φ_{m} ≅ C(I_{m})/C(I_{m}) ∩ C̃_{m}. Using distribution relations it is easy to check that \overline{Cyc}(I_{m})^{σ_{p}−1} ⊂ \overline{C}(I_{m}) ∩ C̃_{m}. By mere rank computation the quotient C(I_{m}) ∩ C̃_{m}/Cyc(I_{m})^{σ_{p}−1} is finite. Let D_{m} be the maximal subfield of F_{m} such that p (totally) splits in D_{m}/ℚ, and let N_{I_{m}/D_{m}} be the norm map. Write \overline{Cyc}(I_{m})[N_{I_{m}/D_{m}}] for the kernel of this norm on \overline{Cyc}(I_{m}). Note that \overline{Cyc}(I_{m})[N_{I_{m}/D_{m}}] ⊂ \overline{C}(I_{m}). Then \overline{Cyc}(I_{m})[N_{I_{m}/D_{m}}] contains Cyc(I_{m})^{σ_{p}−1} with finite index and is maximal with respect to that property inside \overline{C}(I_{m}) because N_{I_{m}/D_{M}} (\overline{Cyc}(I_{m})) ⊂ \overline{Cyc}(D_{m}) is torsion-free. To summarize, we have inclusions of modules with finite index :

\overline{Cyc}(I_{m})^{σ_{p}−1} ⊂ C̃_{m} ∩ \overline{C}(I_{m}) ⊂ \overline{Cyc}(I_{m})[N_{I_{m}/D_{m}}]. (3)

It follows that Tor(Φ_{m}) = \overline{Cyc}(I_{m})[N_{I_{m}/D_{m}}]/\overline{C}(I_{m}) ∩ C̃_{m} and the exact sequence in 1 becomes tautological. Moreover, if we assume that p does not ramify in F and KN_{∞} = 0, then we get C̃_{m}∩\overline{C}(I_{m}) = C̃_{m}∩\overline{C}(F) and n_{U′C} = 0. By theorem 2.10 the sequence of modules (C̃_{m})_{m} satisfies Galois descent and therefore C̃_{m} ∩ \overline{C}(I_{m}) = C̃_{0}. Now it is not difficult to see on a system of generators of \overline{Cyc}(I_{m}) that, in that case, distribution relations imply the equality C̃_{0} = \overline{Cyc}(I_{m})^{σ_{p}−1}, so that 2 follows from 1. □

Using the exact sequence of 1, we see that Tor(Φ_{m}) turns out to be trivial as soon as p does not divide the order of σ_{p}. Intuitively, in the sequence of inclusions (3), any couple of powers of p should occur in some cases as a couple of indices. We prove something far less ambitious which provides the simplest example of non-trivial Tor(Φ_{m}).

Proposition 4.2. — Let F be a real abelian field of degree p such that σ_{p} generates Gal(F/ℚ). Then we have KN_{∞} = 0 and, for all m, isomorphisms Φ_{0} ⟶^{∼} Φ_{m}. Moreover Φ_{0} is finite of order p if and only if at least two distinct rational primes ramify in F and is trivial if and only if a single prime ℓ ≠ p ramifies in F.

Proof. — In the present case we have s_{m}^{+} = 1 for all m, so that Φ_{m} = Tor(Φ_{m}). Then Gal(F/ℚ) is cyclic and p is unramified in F so that KN_{∞} is trivial (see [B1]). By 2 of lemma 4.1 we deduce Φ_{m} = Φ_{0} = Ĥ^{−1}(⟨σ_{p}⟩, \overline{Cyc}(F)). In this case the module Cyc(F) is generated by the single number N_{ℚ(ζ_{f})/F}(1 − ζ_{f}) where f is the conductor of F. If f is a single prime power then this module is Galois free and has trivial cohomology. Else this module is isomorphic to ℤ[ζ_{p}], with residue field 𝔽_{p} as H^{−1}(ℤ/p, ℤ[ζ_{p}]). □

To conclude this section of examples, let us be completely explicit. Take p = 3, ℓ_{1} = 7 and ℓ_{2} = 13. Observe that 3 is not a third power modulo 7 neither modulo 13. Then for i = 1 or 2 the field ℚ(ζ_{ℓ_{i}}) contains a cubic subfield (say F_{i}) which admits 𝔽_{27} as residue field (in other word the ideal (3) remains prime in F_{i}). Let L be the compositum F_{1}F_{2}. Then Gal(L/ℚ) is of type [3, 3] and therefore L admits 4 cubic subfields. These subfields are the inertia subfield at ℓ_{2}, which is F_{1}, the inertia subfield at ℓ_{1}, which is F_{2}, the decomposition subfield at 3, say D, and a fourth subfield F which has conductor f = ℓ_{1}ℓ_{2} and in which the ideal (3) remains prime. In these very simple cases, using proposition 4.2, we have trivial Φ_{0} for the fields F_{i}, an example of finite Φ_{0} of order p for the field F and an example of ℤ_{p}-free Φ_{0} of rank 2 for the field D.

Acknowledgement : Its my pleasure to thank Thong Nguyen Quang Do and Filippo Nuccio for careful reading and helpful comments.

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14th november 2007.

Jean-Robert Belliard, Laboratoire de mathématiques de Besançon UMR 6623, Université de Franche-Comté, 16 route de Gray, 25030 Besançon cedex, FRANCE

    E-mail : jean-robert.belliard@univ-fcomte.fr